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📘 Algebra • Lesson 16 of 22

Algebraic Fractions

Algebraic fractions are fractions that contain letters or algebraic expressions. They follow the same basic rules as ordinary numerical fractions.

✏️ Algebra 📘 Worked examples included 🧠 Step-by-step learning

Learning Objectives

You should be able to:

Introduction

This lesson develops Algebraic Fractions from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

🎬 Cancel factors, not separate terms

6x / 3
→
2x

Cancellation works when numerator and denominator share a factor.

🤔 Think About This...

You already know how to simplify an ordinary fraction.

6/12 = 1/2

We simplify because both 6 and 12 are divisible by the same number.

Algebraic fractions work in the same way.

6x/12

Both the numerator and denominator are divisible by 6.

6x/12 = x/2
Algebraic fractions are ordinary fractions containing algebra.

📖 What is an Algebraic Fraction?

An algebraic fraction is a fraction containing a variable or an algebraic expression in the numerator, denominator or both.

x/5
3x/4
(x + 2)/(x - 1)

The top part is called the numerator.

The bottom part is called the denominator.

The denominator of a fraction can never equal zero.

🚫 Why Can the Denominator Not Be Zero?

Division by zero is undefined.

For example:

5/0

does not have a meaningful value.

Therefore, if we have:

3/(x - 2)

then:

x - 2 ≠ 0
x ≠ 2
The value x = 2 is not allowed because it would make the denominator zero.

🧠 The StudyNest Thinking Method

Before simplifying an algebraic fraction, ask:

👉 What factor is common to the whole numerator and the whole denominator?

Only common factors may be cancelled.

Terms that are being added or subtracted cannot simply be crossed out.

You may cancel factors, but you may not cancel separate terms.

✂️ Simplifying Algebraic Fractions

Example 1

Simplify:

8x/12

🔍 StudyNest Question: What is the highest common factor of 8 and 12?

The highest common factor is 4.

8x ÷ 4 = 2x
12 ÷ 4 = 3
8x/12 = 2x/3

Example 2

Simplify:

15x²/5x

Both the numerator and denominator contain 5x.

15x² ÷ 5x = 3x
5x ÷ 5x = 1
15x²/5x = 3x
This simplification assumes x ≠ 0.

Example 3

Simplify:

6x + 12 ──────── 6

Each term in the numerator is divisible by 6.

(6x + 12)/6
6x/6 + 12/6
x + 2
(6x + 12)/6 = x + 2

Example 4

Simplify:

(x² + 3x)/x

Factorise the numerator first.

x(x + 3)/x

Cancel the common factor x.

(x² + 3x)/x = x + 3
This is valid when x ≠ 0.

⚠ Factors and Terms Are Not the Same

Consider:

(x + 4)/x

You cannot cancel the x because the numerator is a sum of two terms.

(x + 4)/x ≠ 4

But in this expression:

x(x + 4)/x

x is a factor of the whole numerator, so it may be cancelled.

x(x + 4)/x = x + 4
Factorise first. Then look for factors that appear in both the numerator and denominator.

✖️ Multiplying Algebraic Fractions

Multiply the numerators together and multiply the denominators together.

Example 5

Simplify:

(2x/3) × (9/4)

Multiply:

18x/12

Simplify by dividing the numerator and denominator by 6.

(2x/3) × (9/4) = 3x/2

Example 6

Simplify:

(x/5) × (10/x)

The common factor x cancels, provided x ≠ 0.

10/5
(x/5) × (10/x) = 2

➗ Dividing Algebraic Fractions

To divide algebraic fractions, multiply by the reciprocal of the second fraction.

Example 7

Simplify:

(x/4) ÷ (3/2)

Change division into multiplication.

(x/4) × (2/3)
2x/12
x/6
Answer: x/6

➕ Adding Algebraic Fractions

Fractions can only be added directly if they have the same denominator.

Example 8

Simplify:

x/5 + 2/5

The denominators are already the same.

(x + 2)/5
Answer: (x + 2)/5

Example 9

Simplify:

x/6 + x/3

Use a common denominator of 6.

x/6 + 2x/6
3x/6
x/2
Answer: x/2

➖ Subtracting Algebraic Fractions

Example 10

Simplify:

3x/8 − x/8
2x/8
x/4
Answer: x/4

🌍 Real-Life Example

Imagine a cake is shared equally among x people.

Each person receives:

1/x

If another identical cake is shared among the same people, the total each person receives becomes:

1/x + 1/x = 2/x
Fractions involving variables appear naturally whenever quantities are shared.

🎯 Exam Tip

Before cancelling anything, always factorise completely. Most mistakes happen because students cancel terms instead of factors.

📈 Looking Ahead

Some algebraic fractions can also be interpreted graphically using functions. We'll explore these ideas later in the Graphs and Functions topic.

⚠ Common Mistakes

Mistake 1

Cancelling terms instead of factors.

Mistake 2

Adding numerators without first finding a common denominator.

Mistake 3

Forgetting that the denominator cannot equal zero.

Mistake 4

Dividing fractions without using the reciprocal.

Practice Questions

  1. Simplify: 12x/18
  2. Simplify: 20x²/5x
  3. Simplify: (x² + 5x)/x
  4. Multiply: (3x/4) × (8/3)
  5. Divide: (x/5) ÷ (2/5)
  6. Add: x/7 + 2/7
  7. Add: x/4 + x/2
  8. Subtract: 5x/9 − 2x/9
  9. State the value of x that is not allowed in 6/(x−4).
✅ Show Answers
  1. 2x/3
  2. 4x
  3. x + 5
  4. 2x
  5. x/2
  6. (x+2)/7
  7. 3x/4
  8. x/3
  9. x ≠ 4

⭐ What You Can Do Now

🎉 Congratulations!

You have completed the entire Algebra section of StudyNest. Take a moment to celebrate your progress!

From variables... to algebraic fractions... you now have the foundation needed for the rest of Mathematics. 🚀

🧠 Let’s Think Together

Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.

🏆 Algebra Complete!

Lessons completed: 16 of 22

Course Progress 100%

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.

Summary